EXERCISE 6. AVERAGE
IMPORTANT FACTS AND FORMULAE
1. Average
= (sum of observations / number of observations)
2. Suppose
a man covers a certain distance at X kmph and an equal distance at y kmph.
Then, the average speed during the whole journey is (2xy / X + y ) kmph.
SOLVED
EXAMPLES
Ex. 1. Find the average of all
prime number between 30 and 50.
Sol.
there are five prime numbers between 30 and 50.
They are 31,
37, 41 and 47.
Required
average = (31 + 37 + 41 + 43 + 47/5) = 199/5 =39. 8.
Ex. 2. Find the average
of first 40 natural numbers.
Sol. sum of first n natural numbers = n (n + a) / 2.
So, sum of
first 40 natural numbers = 40 x 41 /2 = 820.
Required
average = 820 / 40 = 20.5.
Ex.
3. Find the average of first 20 multiples of 7.
Sol.
required average = 7 ( 1 + 2 + 3 + ….. + 20 ) / 20 = ( 7 x 20 x 21 / 20 x 2) =
(147 / 2 ) = 73.5.
Ex. 4. The average of four consecutive even
numbers is 27. Find the largest of these numbers,
Sol. let the numbers be X,
X + 2, X + 4 and x + 6. Then,
X + (X + 2) + (X
+ 4) + (X +6) /4 = 27 4X + 12 / 4 =
27 X +3 =27 X = 24.
Largest number
= (X + 6) =24 + 6 = 30.
Ex. 5. There are two sections A and B of a class, consisting of 36 and
44 students
respectively. If the average weight of section A is 40 kg and that
of section B is 35 kg, find the average weight of the whole class.
Sol. Total weight of (36 + 44)
students = (36 x 40 + 44 x 35) kg = 2980 kg.
Average weight of
the whole class = (2980 / 80) kg = 37.25 kg.
Ex. 6. Nine persons went to s
hotel for taking their meals. Eight of them spent Rs. 12 each on their earls and the ninth
spent Rs. 8 more than the average expenditure of all the nine. What was the total
money spent by them?
Sol. let the average
expenditure of all the nine be Rs. X.
Then, 12 x 8 +
(X + 8) = 9X or 8X = 104 or X = 13.
Total money
spent = 9X = Rs. (9 x 13) =Rs. 117.
Ex. 7. Of the three
numbers, second is twice the first an is also thrice the third if the average
of the three numbers is 44, find the largest number.
Sol.
let the third number be x. then, second number = 3X. first number =3X / 2
X
+ 3X + 3X / 2= (44 x 3) or 11X/2 = 44 x 3) or 11X / 2= 44 x 3 or x = 24.
So,
largest number = 2nd number =3X = 72.
Ex. 8. The average of 25 results is 18. The
average of first twelve of them is 14 and that of last twelve is 17. Find the
thirteenth result.
Sol. clearly,
thirteenth result = (s of 25 results) – ( s of 24 results)
= (18 x 25) – [(14 x 12) + 1 x 12)]
=
450 – (18 + 204) = 450 – 32 = 8.
Ex. 9. Te average of 11 results is 0. If the
average of first six results is 58 and that of the last six is 63, find the
sixth result.
Sol.
sixth result = (59 x + 63 x 6 – 60 x 11) = 66.
Ex. 10. The average
weight of A, B, C is 45 kg. If the average weight of A and B be 40 kg ad tat of
B and C be43 kg, find the weight of B.
Sol.
let a, B and C represents their individual weights. Then,
A + B +C = (45 x 3) kg = 135 kg.
A
+ B = (40 x 2) kg = 80 kg ad B + C = (43 x 2) kg = 8 kg.
B
= (A + b) + (B + c) – (a + B + C ) = (80 + 8 – 135)kg = 31 kg
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